A Simplified Test for Preference Rationality of
Two-Commodity Choice
Samiran Banerjee and James H. Murphy∗
December 9, 2004
Abstract
We provide a simplified test to determine if choice data from a two-commodity
consumption set satisfies the Generalized Axiom of Revealed Preference (GARP),
and thus the preference or utility maximization hypothesis. We construct an
algorithm for this test and illustrate its application on experimental choice data.
JEL Categories: C91, D11, D12
Keywords: Revealed preference, GARP, laboratory experiments, rationality
∗
Banerjee (corresponding author): Department of Economics, Emory University, Atlanta, GA
30322-2240. Phone: 404-712-8168. Fax: 404-727-4639. Email: sbaner3@emory.edu. Murphy:
College of Business, Western Carolina University, Cullowhee, NC 28723. Phone: 828-227-3731. Fax:
828-227-7414. E-mail: jmurphy@email.wcu.edu. We are grateful to the editor and especially an
anonymous referee for detailed comments that have improved the paper. Additionally, we thank
James Andreoni and Jim Miller for access to their data.
1
Introduction
Much theoretical and applied work in microeconomics presumes that a consumer’s
observed choices result from the maximization of some utility function or preference
relation subject to a binding budget. The centrality of this presumption has motivated attempts to construct and implement falsifiable tests of this ‘utility hypothesis’.
However, applying the appropriate theoretical tests for finite data sets—such as the
Strong Axiom of Revealed Preference (SARP) or the Generalized Axiom of Revealed
Preference (GARP)—to field data presents some difficulties. Varian (1988) showed
that axiomatic tests of the consistency of field data requires the analyst to observe all
(and not just a subset of) the prices and quantities associated with the commodity
space or impose strong behavioral assumptions such as separability.
One way around such problems, especially in the context of controlled experimental methods, is to restrict the dimension of the commodity space. Several studies
which have utilized various axioms of revealed preference to test whether experimental choice data satisfies the preference or utility maximization hypothesis do precisely
this. In an early study, Battalio et al. (1973) analyzed the choices of institutionalized
subjects in a token economy, work later refined and extended by Cox (1997) to incorporate labor supply and portfolio choice. Sippel (1997) elicited university students’
choices from a commodity space of eight common consumption goods. Harbaugh,
Krause and Berry’s (2001) grade school and undergraduate subjects selected from a
two-commodity consumption space (boxes of juice and bags of chips); Andreoni and
Miller’s (2002) subjects chose from bundles consisting of payments to self and to an
anonymous partner.
While the SARP is the appropriate test to determine if a finite set of observed demand choices is consistent with the maximization of a strictly concave utility function
(Matzkin and Richter, 1991), the GARP allows for demand behavior that is somewhat
more general and is consistent with the maximization of a piecewise-linear concave
utility function (Varian, 1982). Checking whether a set of m demand observations
satisfy the SARP or GARP requires the identification of both direct and indirect
chains of revealed preference between any k of the chosen bundles for k = 2, . . . , m.
But when choice is from a two-commodity space, Rose (1958) proved that satisfying
2
the Weak Axiom of Revealed Preference (WARP), which requires only the pairwise
comparison of demanded bundles, is equivalent to satisfying the SARP.
In this paper, we provide an analogous shortcut to check for GARP-consistency
in terms of a Weak Generalized Axiom of Revealed Preference (WGARP) when the
commodity space consists of two goods. Thus a violation of the GARP can be demonstrated in the simple accessible form of a pairwise violation. Section 2 contains definitions and our theoretical result, that choice from a two-commodity consumption
set that is WGARP-consistent is also GARP-consistent. We present an algorithm for
checking the WGARP in Section 3 and apply this algorithm to the data of Andreoni
and Miller (2002). Section 4 concludes.
2
Theoretical Result
n
Let IR+
be a consumption space, where n ≥ 2 indexes the number of commodities.
n
n
; the pair (pi , xi )
chooses a bundle xi ∈ IR+
A subject facing a price vector pi ∈ IR++
constitutes an observation. Let S be a finite set of m observations on a subject, i.e.,
S = {(pi , xi )}m
i=1 .
Definition 1. The set S satisfies the Weak Generalized Axiom of Revealed Preference
(WGARP) if for any two choices xi and xj ,
[pi xi ≥ pi xj ] ⇒ [pj xj ≤ pj xi ].
Following Varian (1982), if xi is chosen when xj is also affordable, we write xi R0 xj ,
where R0 is the binary relation directly revealed preferred to. If xi is chosen when
xj is less costly (i.e., pi xi > pi xj ), we write xi P 0 xj , where P 0 is the binary relation
strictly directly revealed preferred to. Then an alternative statement of the WGARP
is ‘If xi R0 xj , then it is not the case that xj P 0 xi ’.
Definition 2. The set S satisfies the Generalized Axiom of Revealed Preference
(GARP) if for any observation (pi , xi ) and any sequence {(p1 , x1 ), (p2 , x2 ), . . . (pj , xj )}
of length j ≤ m drawn from S, it is the case that
[pi xi ≥ pi x1 & p1 x1 ≥ p1 x2 . . . & pj−1 xj−1 ≥ pj−1 xj ] ⇒ [pj xj ≤ pj xi ].
3
Denoting the transitive closure of the binary relation R0 by R,1 an alternative statement of the GARP is ‘If xi Rxj , then it is not the case that xj P 0 xi ’.
Proposition. When the consumption space contains two commodities (i.e., n = 2),
the WGARP is equivalent to the GARP.
Proof . Sufficiency follows from the definition of GARP. We show necessity by induction on the length of the sequence following Rose (1958). The theorem is trivially true
for any sequence of length one. Suppose it is true for any sequence of length j ≥ 2.
Suppose an observation (pi , xi ) and a sequence {(p1 , x1 ), (p2 , x2 ), . . . , (pj+1 , xj+1 )} is
such that each pair of observations (pi , xi ) and (p1 , x1 ), (p1 , x1 ) and (p2 , x2 ), . . .,
(pj , xj ) and (pj+1 , xj+1 ) satisfies the WGARP, and it is the case that [pi xi ≥ pi x1 & p1 x1 ≥
p1 x2 . . . & pj xj ≥ pj xj+1 ]. We need to show that [pj+1 xj+1 ≤ pj+1 xi ].
For each price vector, normalize the price of the second commodity to unity. Now
if p
j+1
is the smallest or the largest in the sequence, then either (a) pi lies between
pj and pj+1 , or (b) pj lies between pi and pj+1 . If pj+1 is neither the smallest nor the
largest in the sequence, then (c) it lies between two consecutive prices pk and pk+1 ,
k < j. Thus there are three cases to consider:
(a) pi = λpj + (1 − λ)pj+1 for some 0 ≤ λ ≤ 1, or
(b) pj = λpi + (1 − λ)pj+1 for some 0 ≤ λ ≤ 1, or
(c) pj+1 = λpk + (1 − λ)pk+1 for some 0 ≤ λ ≤ 1 and some k < j.
In all cases we need to show that pj+1 (xj+1 − xi ) ≤ 0.
To show this in case (a), note that
pi (xi − xj+1 ) = pi (xi − x1 ) + pi (x1 − xj+1 )
= pi (xi − x1 ) + [λpj + (1 − λ)pj+1 ](x1 − xj+1 )
= pi (xi − x1 ) + λpj (x1 − xj+1 ) + (1 − λ)pj+1 (x1 − xj+1 )
= pi (xi − x1 ) + λpj (x1 − xj ) + λpj (xj − xj+1 ) + (1 − λ)pj+1 (x1 − xj+1 ).
1
For any preference relation R0 on a commodity space X, let R0 denote the set of all possible
transitive extensions of R0 , i.e., for all Ri0 ∈ R0 , (1) Ri0 ⊆ X × X, (2) Ri0 is transitive and (3)
R0 ⊆ Ri0 . The transitive closure of R0 is then the smallest transitive extension of R0 defined as
R = ∩Ri0 ∈R0 Ri0 .
4
Then the first and third terms on the righthand side are non-negative by assumption
and the second and fourth terms are non-negative because of WGARP. Thus pi (xi −
xj+1 ) ≥ 0 and hence pj+1 (xj+1 − xi ) ≤ 0 by WGARP.
For case (b), note that pj (xi − xj+1 ) = pj (xi − xj ) + pj (xj − xj+1 ) is non-negative
because the first term on the righthand side is non-negative by WGARP and the
second by assumption. Therefore, substituting for pj , we get
pj (xi − xj+1 ) = λpi (xi − xj+1 ) + (1 − λ)pj+1 (xi − xj+1 ) ≥ 0.
If pi (xi −xj+1 ) ≤ 0, it follows from the inequality above that pj+1 (xi −xj+1 ) ≥ 0 and we
are done. On the other hand, if pi (xi −xj+1 ) ≥ 0, then by WGARP pj+1 (xj+1 −xi ) ≤ 0.
Finally for case (c), note that
pj+1 (xi − xj+1 ) = pj+1 (xi − xk+1 ) + pj+1 (xk+1 − xj+1 )
= λpk (xi − xk+1 ) + (1 − λ)pk+1 (xi − xk+1 ) + pj+1 (xk+1 − xj+1 )
= λpk (xi − xk ) + λpk (xk − xk+1 ) + (1 − λ)pk+1 (xi − xk+1 ) + pj+1 (xk+1 − xj+1 ).
Then the first, third and fourth terms on the righthand side are non-negative by
WGARP, while the second term is non-negative by assumption. Thus pj+1 (xi −
xj+1 ) ≥ 0, i.e., pj+1 (xj+1 − xi ) ≤ 0 as desired.
Why is it that in the two-commodity case there must be a pairwise violation of
GARP if there is any violation of GARP at all? At the heart of this result lies a crucial
difference between a choice scenario with two versus more than two commodities: in
the case of two goods and a k-wise violation of GARP, the k budget lines can be ranked
from steepest to flattest by normalizing the price of the second good, something not
possible when there are three or more commodities. Then the inequalities in the
k-wise GARP violation can be shown to impose restrictions implying at least one
pairwise GARP violation.
To illustrate this, consider the simplest instance when there are three observations
S = {(p1 , x1 ), (p2 , x2 ), (p3 , x3 )} and where all prices and chosen bundles are distinct.
Suppose that 3-wise GARP is violated: x1 R0 x2 and x2 R0 x3 hold, while x3 P 0 x1 . The
inequalities expressed by these revealed preferred relations are
[p1 x1 ≥ p1 x2 ] & [p2 x2 ≥ p2 x3 ] & [p3 x3 > p3 x1 ].
5
Since we wish to demonstrate that this implies at least one WGARP violation, assume
that WGARP is not violated for any pair of observations from S, i.e., for any pair of
observations i and j from S (i, j = 1, 2, 3, i 6= j),
[pi xi ≥ pi xj ] & [pj xj ≤ pj xi ].
Ranking the budget lines from steepest to flattest, there are six possibilities: (i)
p1 > p2 > p3 , (ii) p1 > p3 > p2 , (iii) p2 > p1 > p3 , (iv) p2 > p3 > p1 , (v) p3 > p1 > p2 ,
and (vi) p3 > p2 > p1 . We analyze the first of these possibilities below.2
In Figure 1, OAD is the budget set from which x1 is chosen, BE the budget line
p2 x1 and CF the budget line p3 x1 . From x3 P 0 x1 (i.e., p3 x3 > p3 x1 ), x3 lies strictly
to the right of CF . However, x3 cannot lie in Ax1 C because that would result in
a WGARP violation between x1 and x3 . Thus x3 must lie strictly to the right of
Ax1 F . Since x1 R0 x2 (i.e., p1 x1 ≥ p1 x2 ), x2 must lie in OAD; however, x2 cannot lie
in Ax1 B because that would result in a WGARP violation between x1 and x2 . Thus
x2 must lie in OBx1 D (excluding the segment Bx1 ). For any x2 in OBx1 D (excluding
segment Bx1 ) and x3 in the area to the right of Ax1 F , one can verify that x2 R0 x3
(i.e., p2 x2 ≥ p2 x3 ) is not possible, a contradiction.
3
Algorithm and Application
Given a data set of m price vectors and m corresponding consumption bundles, we
provide an algorithm to calculate the number of WGARP violations in a given set of
observations.
Let U be an m × m matrix where a typical element uij is defined as
uij =


2 if pi xi > pi xj



1 if pi xi = pi xj



 0 otherwise.
The elements of U then represent the R0 and P 0 relations: uij = 1 iff xi R0 xj but not
xi P 0 xj , and uij = 2 iff xi P 0 xj .
2
A complete analysis of all six cases is available from the authors upon request.
6
Next define another m × m matrix V whose typical element vij is defined as
vij

 u ×u
ij
ji
=
 0
if i < j
otherwise.
Finally define W as an m × m matrix with element wij defined as
wij
Then the data contain
Pm Pm
i=1
j=1

 1 if v ≥ 2
ij
=
0 otherwise.
wij WGARP violations.3 Unlike the GARP algo-
rithm (Varian, 1982 p. 949 and Appendix 2), the WGARP algorithm does not require
computation of the transitive closure of the binary relation R0 .
We apply this WGARP algorithm to Andreoni and Miller’s (2002) data from Sessions 1-4, their base-case scenario.4 In the first two columns of Table 1, we list (by
subject number and reported number of GARP violations) the 13 subjects out of
142 who violated the GARP. Each subject faced eight budgets and selected bundles
consisting of payments to self and to an anonymous partner, a two-good commodity space. In the third column of Table 1 we provide the corresponding number of
WGARP violations calculated using our algorithm. Since Theorem 1 establishes that
WGARP implies GARP, and because GARP-consistent choice can be rationalized by
a piecewise linear utility function which is continuous, concave and monotonic, we
conclude that the WGARP-consistent choice of the remaining 129 subjects is similarly
rationalized.
In columns four and five of Table 1 we reproduce the number of WARP and
SARP violations. Matzkin and Richter (1991, Theorem 2, p. 291) show that SARPconsistent choice can be rationalized by a continuous, strictly concave and strictly
monotonic utility function. Rose’s (1958) result implies that two-commodity WARPconsistent choice is SARP-consistent; consequently, two-commodity WARP-consistent
choice can be rationalized by a continuous, strictly concave and strictly monotonic
utility function. As pointed out by Matzkin and Richter (1991, p. 299), the GARP
implies a weaker notion of rationality than SARP; in particular, GARP-consistent
3
For spreadsheet applications, the algorithm can be shortened by using conditional statements to
replace the V and W matrices. A spreadsheet template is available upon request from the authors.
4
The data can be downloaded from http://www.ssc.wisc.edu/∼andreoni/getdata.html.
7
choice can be rationalized by any constant utility function (which is trivially continuous, concave and monotonic), and by our Theorem 1, the same can be said of
WGARP. However, because the number of WGARP and WARP violations is identical
for the Andreoni and Miller data set,5 the full force of SARP-consistency may be used
to conclude that the 129 subjects behaved as if they were maximizing a continuous,
strictly concave and strictly monotonic utility function.
Table 1 : Violations of Revealed Preference Axioms for Andreoni and Miller Data
Number of Violations
Subject Number
GARP
WGARP
SARP
WARP
3
2
1
3
1
38
7
2
7
2
40
7
3
8
3
41
1
1
1
1
47
1
1
1
1
61
3
1
4
1
72
1
1
1
1
87
1
1
1
1
90
1
1
1
1
104
1
1
2
1
126
1
1
3
1
137
1
1
1
1
139
1
1
1
1
As can be seen from Table 1, there are generally more violations of the GARP than
the WGARP since the former counts both pairwise and k-wise violations (k > 2),
while the latter only counts pairwise violations. To see a concrete instance of this,
consider the choices of subject 3 reproduced in Figure 2. The eight budgets faced by
the subject are labeled from B1 to B8 , while the choices made under each budget are
5
In general, any WARP violation implies an automatic violation of WGARP, but not necessarily
the other way around.
8
shown by the bundles numbered from 1 to 8. These observed choices are summarized
in terms of the revealed preference relations R0 and P 0 in Table 2.
Table 2 : Choices of Subject 3 in Andreoni and Miller Data
Other Bundles Available In Budget
Chosen Bundle
1
1
—
2
3
P
3
4
5
6
0
7
8
0
P0
P0
P0
P
—
R0
P0
—
P0
4
5
2
P0
P0
6
P0
7
P0
8
0
P
R0
R0
R0
—
P0
P0
P0
P0
P0
—
P0
0
0
—
P0
—
R
0
P
P
P0
—
Table 2 shows that bundle 1 was chosen over bundles 2, 7 and 8 under budget
B1 , all of which were less costly; under budget B2 , none of the other bundles chosen
under the other budgets were available when bundle 2 was chosen; bundle 3 was
chosen when bundles 1, 4 and 6 cost the same, while bundles 2, 7 and 8 were less
costly; etc. From the fact that 8R0 3 and 3P 0 8, we have a WGARP violation which
automatically constitutes a GARP violation. From the direct choices 8R0 3 and 3R0 1,
we obtain the indirect relationship 8R1 between bundles 8 and 1; since 1P 0 8, we have
another GARP violation. Note that for this to be a WGARP violation, we would
need either 8R0 1 or 8P 0 1 neither of which is possible since bundle 1 was not available
under budget B8 .
4
Conclusion
The growing use of experimental methods in economics and renewed interest in the
behavioral foundations of choice has led to a number of recent studies testing experimental choice data for consistency with the axioms of revealed preference. The
9
GARP has played a pivotal role in several studies. In this paper, we defined an alternative axiom, the WGARP, which requires only the binary comparison between
chosen bundles. Paralleling Rose’s (1958) findings concerning the equivalence between
the WARP and SARP, we have shown that choice from a two-commodity consumption set that satisfies the WGARP also satisfies the GARP (and vice-versa). Because
checking for GARP-consistency requires an iterative procedure such as Warshall’s
(1962) algorithm to compute the transitive closure of a revealed preference relation,
while WGARP involves only pairwise comparisons of chosen bundles, our finding
offers a conceptually simpler test of whether choices over a two-commodity space
satisfy the hypothesis of preference maximization (as satisfying the GARP implies).
Consequently, the WGARP is particularly valuable in contexts where transparency
in identifying violations of preference rationality is useful, such as classroom demonstrations of the axioms of revealed preference.
Additionally, we have provided some intuition as to why the case of two goods is
special in that any violation of GARP must entail one pairwise violation of GARP.
We have also provided an algorithm to test for WGARP-consistency. On a cautionary note, ours is an equivalence result for WGARP- and GARP-consistency. While
data which contain at least one GARP violation (and hence fails the GARP) must
necessarily fail the WGARP by exhibiting at least one WGARP violation, the total
number of GARP violations need not equal the number of WGARP violations. This
is because each GARP violation is based on direct (i.e., pairwise) and indirect chains
of revealed preference between chosen bundle. Each WGARP violation, on the other
hand, depends only on the direct pairwise comparison between chosen bundles; hence
the number of GARP violations will exceed or equal the number of WGARP violations. Our algorithm is not appropriate if the number of violations rather than their
mere presence or absence is of interest.
References
[1] Andreoni, J. and J. Miller. (2002). “Giving According to GARP: An Experimental Test of the Consistency of Preferences for Altruism.” Econometrica. 70,
737-753.
10
[2] Battalio, R. C. and J. H. Kagel, R. C. Winkler, E. B. Fisher, Jr., R. L. Basmann and L. Krasner. (1973). “A Test of Consumer Demand Theory Using Observations of Individual Consumer Purchases.” Western Economic Journal . 11,
411-428.
[3] Cox, J. C. (1997). “On Testing the Utility Hypothesis.” The Economic Journal .
107, 1054-1078.
[4] Harbaugh, W. T., K. Krause and T. R. Berry. (2001). “GARP for Kids: On
the Development of Rational Choice Behavior.” American Economic Review .
91, 1539-1545.
[5] Matzkin, R. and M. Richter. (1991). “Testing Strictly Concave Rationality.”
Journal of Economic Theory. 53, 287-303.
[6] Rose, H. (1958). “Consistency of Preference: The Two-Commodity Case.” Review of Economic Studies. 25, 124-125.
[7] Sippel, R. (1997). “An Experiment on the Pure Theory of Consumer’s Behaviour.” The Economic Journal . 107, 1431-1444.
[8] Varian, H. R. (1982). “The Nonparametric Approach to Demand Analysis.”
Econometrica. 50, 945-973.
[9] Varian, H. R. (1988). “Revealed Preference with a Subset of Goods.” Journal of
Economic Theory. 46, 179-185.
[10] Warshall, S. (1962). “A Theorem on Boolean Matrices.” Journal of the American
Association of Computing Machinery. 9, 11-12.
11
good 2
A
AD = p1x1
BE = p2x1
CF = p3x1
B
C
x1
O
D
E
F
Figure 1: The Case of Three Observations With p1 > p2 > p3
good 1
150
B6
B4
Payment to other
100
B8
50
B7
4
B1
6
3
5
B3
B2
2
7
50
8
100
Payment to self
Figure 2: Choices of Subject 3
1
B5
150